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Rigid Body Motion and Image Formation . Coordinates of the vector :. 3-D Euclidean Space - Vectors. A “free” vector is defined by a pair of points :. P’. Y’. g. P. y. x. X’. 3D Rotation of Points – Euler angles.

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3 d euclidean space vectors

Coordinates of the vector :

3-D Euclidean Space - Vectors

A “free” vector is defined by a pair

of points :

CS223b, Jana Kosecka

3d rotation of points euler angles

P’

Y’

g

P

y

x

X’

3D Rotation of Points – Euler angles

Rotation around the coordinate axes, counter-clockwise:

z

CS223b, Jana Kosecka

rotation matrices in 3d
Rotation Matrices in 3D
  • 3 by 3 matrices
  • 9 parameters – only three degrees of freedom
  • Representations – either three Euler angles
  • or axis and angle representation
  • Properties of rotation matrices (constraints between the

elements)

CS223b, Jana Kosecka

rotation matrices in 3d5
Rotation Matrices in 3D
  • 3 by 3 matrices
  • 9 parameters – only three degrees of freedom
  • Representations – either three Euler angles
  • or axis and angle representation
  • Properties of rotation matrices (constraints between the

elements)

Columns are orthonormal

CS223b, Jana Kosecka

canonical coordinates for rotation
Canonical Coordinates for Rotation

Property of R

Taking derivative

Skew symmetric matrix property

By algebra

By solution to ODE

CS223b, Jana Kosecka

3d rotation axis angle
3D Rotation (axis & angle)

Solution to the ODE

with

or

CS223b, Jana Kosecka

rotation matrices
Rotation Matrices

Given

How to compute angle and axis

CS223b, Jana Kosecka

3d translation of points

P’

t

Y’

x

x’

P

z’

y

z

3D Translation of Points

Translate by a vector

CS223b, Jana Kosecka

rigid body motion homogeneous coordinates

Homogeneous coordinates:

Rigid Body Motion – Homogeneous Coordinates

3-D coordinates are related by:

Homogeneous coordinates are related by:

CS223b, Jana Kosecka

rigid body motion homogeneous coordinates11

Homogeneous coordinates:

Rigid Body Motion – Homogeneous Coordinates

3-D coordinates are related by:

Homogeneous coordinates are related by:

CS223b, Jana Kosecka

properties of rigid body motions
Properties of Rigid Body Motions

Rigid body motion composition

Rigid body motion inverse

Rigid body motion acting on vectors

Vectors are only affected by rotation – 4th homogeneous coordinate is zero

CS223b, Jana Kosecka

rigid body transformation
Rigid Body Transformation

Coordinates are related by:

Camera pose is specified by:

CS223b, Jana Kosecka

rigid body motion continuous case
Rigid Body Motion - continuous case
  • Camera is moving
  • Notion of a twist
  • Relationship between velocities

CS223b, Jana Kosecka

image formation
Image Formation
  • If the object is our lens the refracted light causes the images
  • How to integrate the information from all the

rays being reflected from the single point

on the surface ?

  • Depending in their angle of incidence, some are

more refracted then others – refracted rays all

meet at the point – basic principles of lenses

  • Also light from different surface points may hit the same lens point but they are refracted differently - Kepler’s

retinal theory

CS223b, Jana Kosecka

thin lens equation
Thin lens equation
  • Idea – all the rays entering the lens parallel to the optical axis on one side, intersect on the other side at the point.

Optical axis

f

f

CS223b, Jana Kosecka

lens equation
Lens equation

p

O

z’

f

f

Z’

Z

z

  • distance behind the lens at which points becomes in
  • focus depends on the distance of the point from the lens
  • in real camera lenses, there is a range of points which
  • are brought into focus at the same distance
  • depth of field of the lens , as Z gets large – z’ approaches f
  • human eye – power of accommodation – changing f

CS223b, Jana Kosecka

image formation perspective projection
Image Formation – Perspective Projection

“The School of Athens,” Raphael, 1518

CS223b, Jana Kosecka

pinhole camera model
Pinhole Camera Model

Pinhole

Frontal

pinhole

CS223b, Jana Kosecka

more on homogeneous coordinates
More on homogeneous coordinates

In homogenous coordinates – these represent the

Same point in 3D

The first coordinates can be obtained from the

second by division by W

What if W is zero ?

Special point – point at infinity – more later

In homogeneous coordinates – there

is a difference between point and vector

CS223b, Jana Kosecka

pinhole camera model21
Pinhole Camera Model
  • Image coordinates are nonlinear function of world coordinates
  • Relationship between coordinates in the camera frame and sensor plane

2-D coordinates

Homogeneous coordinates

CS223b, Jana Kosecka

image coordinates

metric

coordinates

Linear transformation

pixel

coordinates

Image Coordinates
  • Relationship between coordinates in the sensor plane and image

CS223b, Jana Kosecka

calibration matrix and camera model

Calibration matrix

(intrinsic parameters)

Projection matrix

Camera model

Calibration Matrix and Camera Model
  • Relationship between coordinates in the camera frame and image

Pinhole camera

Pixel coordinates

CS223b, Jana Kosecka

calibration matrix and camera model24
Calibration Matrix and Camera Model
  • Relationship between coordinates in the world frame and image

Pinhole camera

Pixel coordinates

More compactly

Transformation between camera coordinate

Systems and world coordinate system

CS223b, Jana Kosecka

radial distortion
Radial Distortion

Nonlinear transformation along the radial direction

New coordinates

Distortion correction: make lines straight

Coordinates of distorted points

CS223b, Jana Kosecka

image of a point
Image of a point

Homogeneous coordinates of a 3-D point

Homogeneous coordinates of its 2-D image

Projection of a 3-D point to an image plane

CS223b, Jana Kosecka

image of a line homogeneous representation
Image of a line – homogeneous representation

Homogeneous representation of a 3-D line

Homogeneous representation of its 2-D image

Projection of a 3-D line to an image plane

CS223b, Jana Kosecka

image of a line 2d representations
Image of a line – 2D representations

Representation of a 3-D line

Projection of a line - line in the image plane

Special cases – parallel to the image plane, perpendicular

When  -> infinity - vanishing points

In art – 1-point perspective, 2-point perspective, 3-point perspective

CS223b, Jana Kosecka

vanishing points
Vanishing points

Different sets of parallel lines in a plane intersect

at vanishing points, vanishing points form a horizon line

CS223b, Jana Kosecka

ames room illusions
Ames Room Illusions

CS223b, Jana Kosecka

more illusions
More Illusions

Which of the two monsters is bigger ?

CS223b, Jana Kosecka